4 S ep 1 99 7 Broad Histogram Monte Carlo

We propose a new Monte Carlo technique in which the de-generacy of energy states is obtained with a Markovian process analogous to that of Metropolis used currently in canonical simulations. The obtained histograms are much broader than those of the canonical histogram technique studied by Ferrenberg and Swendsen. Thus we can reliably reconstruct ther-modynamic functions over a much larger temperature scale also away from the critical point. We show for the two-dimensional Ising model how our new method reproduces exact results more accurately and using less computer time than the conventional histogram method. We also show data in three dimensions for the Ising ferromagnet and the Edwards Anderson spin glass. In the simulation of thermal systems one typically needs to calculate a variable < Q > T as a function of temperature T. Usually one would have to perform independent Monte Carlo simulations at different values of temperature. An appealing strategy to avoid such multiple simulations is the canonical histogram method [1,2]: The histogram P T (E) of the energies at one given temperature T 0 is measured and then the distribution at a different temperature T is obtained by reweighting, i.e. by multiplying with exp(E/T − E/T 0) and normalizing. In order to obtain the average < Q > T , one needs to accumulate also the histogram of Q as a function of energy E. The thermal average at temperature T is then (1) where < Q(E) > means the average value of Q obtained at fixed energy E. For simplicity we set k B = 1 in this article. The histogram method has been carefully checked on various models [3]. Its main disadvantage is that the canonical distribution P T (E) of the energy is rather narrowly peaked around the average value < E > T 0 (and the more so the larger the system) so that when T is not close to T 0 the reweighting factor is very small there and very large near < E > T. One also has strong statistical fluctuations stemming from the tails of the distributions. So, in order to get reliable results the tails must be sampled very well by making good statistics and the temperatures considered should not be far from T 0. We want to introduce here a completely different technique based on the calculation of the degeneracy g(E) of energy states from histograms of adequate macroscopic …